Similarity & transformations
The idea
Similar figures share a shape but not necessarily a size: one is the image of the other under a dilation — a uniform scaling from a center point — possibly combined with rigid motions. All corresponding angles are equal and all corresponding sides share one scale factor k. For triangles, the angle–angle (AA) criterion does the heavy lifting: two matching angles force the third to match, since a triangle's angles always total 180°, so the triangles must be scaled copies. Similarity powers indirect measurement, maps, and the very definition of the trigonometric ratios.
Working a similarity problem is disciplined bookkeeping: establish similarity (usually by AA), write the correspondence of vertices in order, then set up ratios between whole corresponding sides. The notorious trap appears when a line parallel to one side cuts across a triangle: the small triangle's side must be compared with the large triangle's WHOLE side, never with the leftover piece. Remember also that areas scale by k², not k — doubling every length quadruples the area — so a scale factor must be squared before it touches an area.
Worked example
In triangle ABC, point D lies on side AB and point E lies on side AC, with DE parallel to BC. Given AD = 6, DB = 4, and DE = 9, find BC.
- Establish similarity: DE parallel to BC makes angle ADE equal to angle ABC (corresponding angles), and angle A is shared, so triangle ADE is similar to triangle ABC by AA.
- Find the scale factor from whole corresponding sides measured from vertex A: AD/AB = 6/(6 + 4) = 6/10 = 3/5. Using AD/DB = 6/4 here is the classic error — DB is a leftover piece, not a side of either triangle.
- Apply the ratio to the wanted pair: DE/BC = 3/5, so BC = 9 × 5/3 = 15.
- Verify the proportion: 9/15 reduces to 3/5, matching 6/10 — the small triangle really is a 3/5-scale copy of the large one.
Answer. BC = 15, because triangle ADE is a 3/5-scale copy of triangle ABC.
Check your understanding
- Why do two matching angles force triangles to be similar, while two matching sides alone force nothing at all?
- What distinguishes the ratio AD/AB from AD/DB in the parallel-line setup, and why does only one belong in the similarity proportion?
- If the scale factor between two similar figures is 3, what happens to their perimeters and their areas, and why do the two answers differ?
- How does similarity justify defining sine and cosine as ratios that depend only on the angle, never on the triangle's size?
Build the foundations first
Similarity & transformations builds on these concepts. If any feel shaky, start there.